
where \(\varphi _{i,l,k} \triangleq \sin (\theta _{i,l}) \sin (\phi _{i,l}) - \sin (\theta _{i,k}) \sin (\phi _{i,k})\) . We know, however, that the submatrices included on the diagonal of \({\mathbf {C}}_i\)  are related to the autocorrelation matrix of \(\sum _{l = 1}^{L_i}e^{j\pi (m-1) \sin (\theta _{i,l}) \sin (\phi _{i,l})}\gamma _{i,l}\alpha _z(\phi _{i,l})\) , which is the linear combination of the array response vectors \(\alpha _z(\phi _{i,l})\)  for \(l = 1,2,\ldots ,L_i\) . We calculate \({\mathbf {C}}_{i,z} = \sum _{m = 1}^{M_{i,y}} [{\mathbf {C}}_{i}]_{mm}\) , based on which
we can apply the root MUSIC algorithm [1]} for estimating \(\phi _{i,1}, \ldots , \phi _{i,L_i}\) .In this paper, we assume that the gain of the LoS path is much larger than that of any of the NLoS paths. In this way, it is highly probable to find the correct LoS AoA without encountering any path association/mismatch issue. In practice, we replace \({\mathbf {C}}_{i}\)  with \(\hat{{\mathbf {C}}}_{i}\)  for performing the estimation. The calculation of
 (REF ) involves \(M_{i,y} L_i^2 M^2_{i,z}\)  addition operations and \(M_{i,y} L_i^2 M^2_{i,z}\)  multiplication operations. In addition, the computational complexity for root MUSIC here is \(O\big (M_{i,z}^{3}\big )\) .
